Consider an asset whose price follows the geometric Brownian motion process

where \(z\) is a standard Wiener process.

(a) At time \(t\) (when \(S(t)\) is known), what is the expected value of the asset's price at the future time \(T\) ? This is denoted \(\mathrm{E}_{t}[S(T)]\).

(b) Let \(W(t)=\mathrm{E}_{t}[S(T)]\). What is the process governing \(W(t)\) ?

(c) Let \(W(0)=1\). What are \(\mathrm{E}_{0}[W(t)]\) and \(\mathrm{E}_{0}[\ln W(t)]\) ?

(d) What are \(\operatorname{Var}_{0}[W(t)]\) and \(\operatorname{Var} 0[\ln W(t)]\) ?

Transcribed Image Text:

ds(t)=us(t) dt +S(t) dz,